On the Minimal Positive Homothetic Image of a Simplex Containing a Convex Body

Let $C$ be a convex body, and let $S$ be a nondegenerate simplex in $\mathbb R^n$. It is proved that the minimal coefficient $\sigma>0$ for which the translate of $\sigma S$ contains $C$ is
$$ \sum_{j=1}^{n+1}\max_{x\in C}(-\lambda_j(x))+1, $$
where $\lambda_1(x),\dots,\lambda_{n+1}(x)$ are the barycentric coordinates of the point $x\in\mathbb R^n$ with respect to $S$. In the case $C=[0,1]^n$, this quantity is reduced to the form $\sum_{i=1}^n 1/d_i(S)$, where $d_i(S)$ is the $i$th axial diameter of $S$, i.e., the maximal length of the segment from $S$ parallel to the $i$th coordinate axis.